PROP. V. THEOR. Pyramids of the same altitude which have triangular See N. bases, are to one another as their bases. Let the pyramids of which the triangles ABC,DEF are the bases, and of which the vertices are the points. G, H, be of the same altitude: as the base ABC to the base DEF, so shall the pyramid ABCG be to the pyramid DEFH. For, if it be not so, the base ABC must be to the -base DEF, as the pyramid ABCG to a solid either less than the pyramid DEFH, or greater than it ‡. First, let it be to a solid less than it, viz. to the solid Q: and divide the pyramid DEFH into two equal pyramids, similar to the whole, and into two equal prisms; therefore these two prisms are greater than the half of the * 3. 12. whole pyramid. And again, let the pyramids made by * * 4. 12. this division be in like manner divided, and so on† † Lemma until the pyramids which remain undivided in the 1. 12. pyramid DEFH be, all of them together, less than the excess of the pyramid DEFH above the solid Q: let these, for example be the pyramids DPRS, STYH: therefore the prisms, which make the rest of the pyramid DEFH, are greater than the solid Q. Divide likewise the pyramid ABCG in the same manner, and into as many parts, as the pyramid DEFH. Therefore as the base ABC to the base DEF, so* are the prisms in the pyramid ABCG to the prisms in the pyramid DEFH: but as the base ABC to the base DEF, so by hypothesis, is the pyramid ABCG to the solid Q; and therefore, as the pyramid ABCG to the solid Q, so are the prisms in the pyramid ABCG to the prisms in the pyramid DEFH; but the pyramid ABCG is greater than the prisms contained in it; wherefore also the solid Q is greater than the *14. 5. prisms in the pyramid DEFH: but it is also less, which is impossible. Therefore the base ABC is not to the base DEF, as the pyramid ABCG to any solid which is less than the pyramid DEFH. In the same manner it may be demonstrated, that the base DEF is not to the base ABC, as the pyramid DEFH to any solid which is less than the pyramid ABCG. Nor can This may be explained in the same way as at the notet in Proposition 2, in the like case. *14. 5. the base ABC be to the base DEF, as the pyramid * B M N T H mid ABCG, because the solid Z is greater than the pyramid DEFH; and therefore, as the base DEF to the base ABC, so is the pyramid DEFH to a solid less than the pyramid ABCG; the contrary to which has been proved: therefore the base ABC is not to the base DEF, as the pyramid ABCG to any solid which is greater than the pyramid DEFH. And it has been proved that neither is the base ABC to the base DEF, as the pyramid ABCG to any solid which is less than the pyramid DEFH. Therefore, as the base ABC is to the base DEF, so is the pyramid ABCG to the pyramid DEFH. Wherefore, pyramids, &c. Q. E. D. See N. PROP. VI. THEOR. Pyramids of the same altitude which have polygons for their bases, are to one another as their bases. Let the pyramids which have the polygons ABCDE, FGHKL, for their bases, and their vertices in the points M, N, be of the same altitude: as the base ABCDE to the base FGHKL, so shall the pyramid ABCDEM be to the pyramid FGHKLN. Divide the base ABCDE into the triangles ABC, ACD, ADE, and the base FGHKL into the triangles FGH, FHK, FKL: and upon the bases ABC, ACD, ADE, let there be as many pyramids of which the common vertex is the point M, and upon the remain This may be explained the same way as the like at the markt in Prop. 2. * are all * A B M DI * 2 Cor. 24. 5. ing bases as many pyramids having their common ver- PROP. VII. THEOR. Every prism having a triangular base may be divided into three pyramids that have triangular bases, and are equal to one another. Let there be a prism of which the base is the triangle ABC, and DEF the triangle opposite to it: the prism ABCDEF may be divided into three equal pyramids having triangular bases. * Join BD, EC, CD: and because ABED is a parallelogram of which BD is the diameter, the triangle ABD is equal to the triangle EBD; therefore the py- * 34. 1. ramid of which the base is the triangle ABD and vertex the point C, is equal to the pyramid of which the base is the triangle EBD, and vertex the point C: but this pyramid is the same with the pyramid the base of which is the triangle EBC, and vertex the point D ; * 5. 12. * 34. 1. 6. 12. D A F B for they are contained by the same planes: therefore the COR. 1. From this it is manifest, that every pyramid is the third part of a prism which has the the same base, and is of an equal altitude with it: for if the base of the prism be any other figure than a triangle, it may be divided into prisms having triangular bases. COR. 2. Prisms of equal altitudes are to one another as their bases; because the pyramids upon the same. bases, and of the same altitude, are * to one another as their bases. PROP. VIII. THEOR. Similar pyramids, having triangular bases, are one to another in the triplicate ratio of that of their homologous sides. Let the pyramids having the triangles ABC, DEF, for their bases, and the points G, H for their vertices, be similar and similarly situated: the pyramid ABCG shall have to the pyramid DEFH, the triplicate ratio of that which the side BC has to the homologous side EF. * Complete the parallelograms ABCM, GBCN, ABGK, and the solid parallelopiped BGML, coutained by these planes and those opposite to them: and, in like manner, complete the solid parallelopiped EHPO contained by the three parallelograms DEFP, HEFR, DEHX, and those opposite to them. And because the pyramid ABCG is similar to the pyramid DEFH the angle ABC is equal to the 11 Def. angle DEF, and the angle GBC to the angle HEF, and 11. ABG to DEH: and AB is to BC, as DE to EF; that is the sides about the equal angles are proportionals: wherefore the parallelogram BM is similar to EP: for the same reason, the parallelogram BN is similar to ER, and BK to EX: therefore the three parallelograms BM, BN, * K L R B * * 11. 1. Def. 6. * 33. 11. BK, are similar to the three EP, ER, EX: but the three BM, BN, BK, are equal and similar to the three * 24. 11. which are opposite to them, and the three EP, ER, EX, equal and similar to the three opposite to them: wherefore the solids BGML, EHPO are contained by the same number of similar planes: and their solid angles are B. 11. equal; and therefore the solid BGML is similar to * 11 Def. the solid EHPO: but similar solid parallelopipeds have the triplicate ratio of that which their homologous sides have; therefore the solid BGML has to the solid EHPO the triplicate ratio of that which the side BC has to the homologous side EF: but as the solid BGML is to the solid EHPO, so is the pyramid *15.5. ABCG to the pyramid DEFH; because the pyramids are the sixth part of the solids, since the prism, which is the half of the solid parallelopiped, is triple of the pyramid: wherefore likewise the pyramid ABCG has to the pyramid DEFH, the triplicate ratio of that which BC has to the homologous side EF. Q. E. D. * COR. From this it is evident, that similar pyramids * 28. 11. *7. 12. |
